Short course Multi-trait
LAPEA

Outline

Background

Correlation between traits

Genetic correlation

• Greatest interest for breeders

• Positive or negative and favorable or unfavorable, depending on goals. Ex 1: plant height and flowering date in barley are positively correlated. This correlation is favorable if harvest grains is the goal (short and early-maturing). But for forage (tall and late-maturing) is unfavorable (Bernardo et al. 2002).

• Causes

  • Linkage: genes located on the same chromosome are genetically linked and do not segregate independently, like genes located on different chromosomes (can be dissipated by cycles of meiosis).

  • Pleiotropy: two traits are controlled by the same gene (can not be dissipated, more permanent).

Response to selection

The mean genetic change in the trait of interest in a population.

Direct response to selection

The selection criteria is based on the trait(s) of interest in the target environment.

\[ Rx = {k}_{p} {h}_{x}{\sigma_a}x\] \[Ry = {k}_{p} {h}_{y}{\sigma_a}y\]

\({k}_{p}\): is the selection differential when the proportion of selected individuals is \(p\);
\({h}\): is the accuracy of individual selection (direct selection);
\({\sigma_a}\): is the standard deviation of breeding values.

Indirect selection

• The criteria is based on trait(s) that may be associated with the trait(s) of interest (secondary traits). Ex: number of stalks and tons of cane per hectare. Also, the same trait measured in different environments(Rutkoski 2019):

Correlated response to selection

• Correlated response to selection: selection for one trait will cause a correlated response in the other (if genetic correlation exists). The expected response of a trait Y, when selection is applied to another trait X:

\[CR_{Y} = {k}_{p} {h}_{x} {h}_{y}{r}_{g}\sigma_{p_{Y}}\] - Indirect selection is expected to be better than direct selection if the secondary trait has substantially higher heritability, and the genetic correlation is high.

• Indirect selection can also be advantageous if much larger population sizes are possible compared to direct selection. Also, if secondary trait is easier and/or cheaper to evaluate, and measurable on both sex for example.

Tandem selection

set.seed(1234)
x <- rnorm(200, mean = 10, sd = 4)
y <- rnorm(200, mean = 6, sd = 2)

dat <- data.frame(x,y)

# Visualize the scatter plot of traits A and B
plot(x,y, pch=19, ylim = c(min(dat$y), max(dat$y)),xlim = c(min(dat$x), max(dat$x)),rect(par("usr")[1], par("usr")[3],
          par("usr")[2], par("usr")[4], col = "lightgrey"),
     cex.lab=1.5, cex.axis=1.8,
     xlab="Trait A", ylab="Trait B",col=ifelse(y >5, 'red', 'blue'))
title(main = "SELECT FOR TRAIT B FOR SEVERAL CYCLES",  cex.main = 2.5, col.main= "darkgreen")
abline(h=5, col="black", lwd = 2, lty = 2)

set.seed(1234)
x <- rnorm(200, mean = 10, sd = 4)
y <- rnorm(200, mean = 6, sd = 2)

dat <- data.frame(x,y)

# Visualize the scatter plot of traits A and B
plot(x,y, pch=19, ylim = c(min(dat$y), max(dat$y)),xlim = c(min(dat$x), max(dat$x)),
     rect(par("usr")[1], par("usr")[3],
          par("usr")[2], par("usr")[4], col = "lightgrey"),
     cex.lab=1.5, cex.axis=1.8,
     xlab="Trait A", ylab="Trait B",col=ifelse(x > 10, 'red', 'blue'))
title(main = "SELECT FOR TRAIT A FOR SEVERAL CYCLES",  cex.main = 2.5, col.main= "darkgreen")
abline(v=10, col="black", lwd = 2, lty = 2)

Independent culling levels

A certain level of merit (minimun level of performance) is established for each trait, and all individuals below that level are discarded regardless of values for other traits.

set.seed(1234)
x <- rnorm(200, mean = 10, sd = 4)
y <- rnorm(200, mean = 6, sd = 2)

dat <- data.frame(x,y)

# Visualize the scatter plot of traits A and B
plot(x,y, pch=19, ylim = c(min(dat$y), max(dat$y)),xlim = c(min(dat$x), max(dat$x)),
     rect(par("usr")[1], par("usr")[3],
          par("usr")[2], par("usr")[4], col = "lightgrey"),
     cex.lab=1.5, cex.axis=1.8,
     xlab="Trait A", ylab="Trait B",col=ifelse(x > 10 & y >5, 'red', 'blue'))
abline(v=10, h=5, col="black", lwd = 2, lty = 2)

Index selection

Select for \(n\) traits simultaneously by using some index of net merit:

\[I = b_{1}X_{1} + b_{1}X_{1} + ... + b_{n}X_{n}\] \(b_{1}\) is the weight for trait \(i\) and \(X_{1}\) is the phenotypic value for trait \(i\).

Example adapted from (Bernardo et al. 2002):
Price of 60 kg bag of maize: R$ 90,00
To dry a 60 kg bag of maize: R$ 0,25
Target moisture concentration: 13%

A new candidate maize hybrid genotype has a yield of 100 bags of 60kg/ha and has a moisture of 15%. Then, the profit can be expressed in the form of the following selection index:

\[I = (90)(Yield) - (0,25)(Moisture - 13\%)(Yield)\]

Uni-trait model

The BLUP methodology uses mixed models for the genetic analysis and provides accurate and least biased prediction of breeding values.

Mixed model formulation

\[ y = \mathbf{X}b + \mathbf{Z}u + e \] \(y\): is an (nx1) vector of observations (phenotypes)
\(b\): is an (px1) vector with fixed effects (e.g. environment)
\(u\): is an (qx1) vector with random effects of breeding values; \(u \sim N(\mathbf{0},\mathbf{K} \sigma_{g}^2)\)
\(e\): is an (nx1) vector with random effects of residuals; \(e \sim N(\mathbf{0},\mathbf{R} \sigma_{e}^2)\)
\(\mathbf{X}\) and \(\mathbf{Z}\): are design matrices relating obs to fixed and random effects, respectively.

Uni-trait model

\[ y = \mathbf{X}b + \mathbf{Z}u + e \] \[\begin{align} \begin{bmatrix} u \\ e \\ \end{bmatrix} \sim MVN \begin{pmatrix} \begin{bmatrix} 0 \\ 0 \\ \end{bmatrix} , \begin{bmatrix} \sigma_{g}^2 \mathbf{K} & 0 \\ 0 & \sigma_{e}^2 \mathbf{R} \\ \end{bmatrix} \end{pmatrix} \end{align}\]

Advantages

• Allows the analysis of unbalanced data

• Exploit information from relatives

Furthermore, it can be extended to incorporate/account for more effects, such as:

• Rows, columns, environments, permanent environmental…

• Interaction terms (e.g GXE)

• Covariance and correlation structures (relationship matrices and Correlation matrices)

• Heterogeneous variances

• Correlated traits

Multi-trait model

\[\begin{gather*} y_{i} = X_{i}b_{i} + Z_{i}u_{i} + e_{i} , & i = 1, 2,.. trait \end{gather*}\]

\[\begin{align} \begin{bmatrix} y_{1} \\ y_{2} \\ \end{bmatrix} = \begin{bmatrix} X_{1} & 0 \\ 0 & X_{2} \\ \end{bmatrix} \begin{bmatrix} b_{1} \\ b_{2} \\ \end{bmatrix} + \begin{bmatrix} Z_{1} & 0 \\ 0 & Z_{2} \\ \end{bmatrix}\begin{bmatrix} u_{1} \\ u_{2} \\ \end{bmatrix} + \begin{bmatrix} e_{1} \\ e_{2} \\ \end{bmatrix} \end{align}\]

\[\begin{align} \begin{bmatrix} u \\ e \\ \end{bmatrix} \sim MVN \begin{pmatrix} \begin{bmatrix} \mathbf{G}\otimes\mathbf{K} & 0 \\ 0 & \mathbf{\Sigma}\otimes{I} \\ \end{bmatrix} \end{pmatrix} \end{align}\]

Advantages

• The prediction accuracy of low-heritability, difficult, and/or expensive to measure traits can be increased by using multi-trait models when the degree of correlation between traits is at least moderate, to improve the target trait or all the correlated traits simultaneously (Calus and Veerkamp 2011).

• Multi-trait models can be useful for increasing prediction accuracy when the traits of interest are not measured in the individuals of the testing set, but were observed in individuals in the training set (Jia and Jannink 2012).

Hands-on using R

 Installing packages
 
 # Github version
 install.packages('devtools');
 library(devtools);
 install_github('covaruber/sommer')
 
 # or
 
 # CRAN version
 install.packages('sommer',dependencies = TRUE))
 library(sommer)

Hands-on using R

# rrBLUB (single-trait and no marker information used)

# GBLUP (single-trait with marker information used)

################################################################################
################################################################################
# Cross-validation Single-trait models (rrBLUP) using sommer
# Datasets
data(DT_cpdata)
# Phenotypic data
DT <- DT_cpdata
head(DT)
# Removing rows with NAS because is preventing correlation in corss validation 
DT <- na.omit(DT)
# Marker data
GT <- GT_cpdata
GT_cpdata[1:5,1:5]
# Chromossome positions
#MP <- MP_cpdata
#head(MP_cpdata)
# Genomic relationship matrix
A <- A.mat(GT)
colnames(A) <- rownames(A) <- rownames(DT)
A[1:5,1:5]
# Creating new dataset
data.1.2 <- DT
# Scaling phenotypes
data.1.2$Yield <- as.vector(scale(data.1.2$Yield, center = T, scale = T)) # Yield
data.1.2$FruitAver <- as.vector(scale(data.1.2$FruitAver, center = T, scale = T)) # fruit avarage
data.1.2$Firmness <- as.vector(scale(data.1.2$Firmness, center = T, scale = T)) # firmness
data.1.2$color <- as.vector(scale(data.1.2$color, center = T, scale = T)) # color
######################################################################################
# Cross-validation Single-trait models (GBLUP) using sommer
#30 Replicates
rep <- 3
#5 random partitions
fold <- 5
n <- dim(data.1.2)[1]
Replicates <- data.frame(Replicate = 1:rep,MSEP = NA,Cor = NA) 
for (j in 1:rep){ 
  set.seed(j)
  Partitions <- replicate(fold,sample(n,0.20*n,replace = F))
  Table <- data.frame(Partitions = 1:fold,MSEP = NA,Cor = NA)
  for(i in 1:fold) {
    tst <- Partitions[,i]
    data.1.2$y_NA <- data.1.2$Yield
    data.1.2$y_NA[tst] <- NA
    #M1
    fit = mmer(y_NA ~ 1,
         random=~vsr(id,Gu=A) + vsr(Rowf),
         rcov=~units,
         data=data.1.2,verbose = FALSE)
    #BLUPs
    bv <- fit$U$`u:id`$y_NA
    #Prediction of testing
    yp_tst <- bv[tst]
    #MSEP and Cor
    Table$MSEP[i] <- mean((data.1.2$Yield[tst]-yp_tst)^2)
    Table$Cor[i] <- cor(data.1.2$Yield[tst],yp_tst)
  }
  Replicates$MSEP[j] <- mean((na.omit(Table$MSEP))) # mean of folds
  Replicates$Cor[j] <-mean((na.omit(Table$Cor))) # mean of folds
}

mean(Replicates$MSEP) # mean of replicates
mean(Replicates$Cor) # mean of replicates
######################################################################################

# rrBLUP (multi-trait and no marker information used)

# GBLUP (multi-trait with marker information used)